Connectivity Analysis in AFNI

Gang ChenSSCC/NIMH/NIH/HHS

105/06/23

File: Connectivity.pdf

Structure of this lecture Two categories of connectivity analysis

Seed-based (vs. functional connectivity) Network-based (vs. effective connectivity)

Seed-based analysis Simple correlation Context-dependent correlation (PPI) Seed-based bivariate autoregression

Network-based analysis Structural equation modeling (SEM) Vector autoregression (VAR) (aka Granger

causality) Structural vector autogression (SVAR)

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Overview: Connectivity analysis Typical FMRI data analysis

Massively univariate (voxel-wise) regression: y = Xβ+ε Relatively robust and reliable May infer regions involved in a task/state, but can’t

say much about the details of a network Network analysis

Informationo Seed region, some or all regions in a networko Neuroimaging data (FMRI, MEG, EEG): regional time series

Infer interregional communicationso Inverse problem: infer neural processes from BOLD signalo Based on response similarity (and sequence)

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Overview: Connectivity analysis Two types of network analysis

Not sure about ALL the regions involvedo Seed-based: use a seed region to search for other ROIs

If all regions in a network knowno Prior knowledgeo Network-based: A network with all relevant regions

knowno Everything is relative: No network is fully self-

contained Data types

Mainly FMRI Some methodologies may work for MEG, EEG Not for DTI

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Seed-based analysis: ROI search Regions involved in a network are unknown

Bi-regional (seed vs. whole brain) (3d*): brain volume as input

Mainly for ROI search Popular name: functional connectivity Basic, coarse, exploratory with weak assumptions Methodologies: simple correlation, PPI, bivariate

autoregression Weak interpretation: may or may not indicate

directionality/causality

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Network-based analysis Regions in a network are known

Multi-regional (1d*): ROI data as input Model validation, connectivity strength testing Popular name: effective or structural

connectivity Strong assumptions: specific, but with high risk Methodologies: SEM, VAR, SVAR, DCM Directionality, causality (?)

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Common Preparatory Steps Warp brain to standard space

adwarp, @auto-tlrc, align_epi_anat.py Create ROI

Peak voxel Sphere around a peak activation voxel: 3dUndump –master … –srad … Activation cluster-based (biased unless from independent data?) Anatomical database Manual drawing

Extract ROI time series Average over ROI: 3dmaskave –quiet –mask, or 3dROIstats -quiet –mask Principal component among voxels within ROI: 3dmaskdump, then 1dsvd Seed voxel with peak activation: 3dmaskdump -noijk -dbox

Remove effects of no interest 3dSynthesize and 3dcalc 3dDetrend –polort RETROICORR/RetroTS.m 3dBandpass

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Simple Correlation Analysis Seed vs. rest of brain ROI search based on response similarity

Looking for regions with similar signal to seed Correlation at individual subject level

Usually have to control for effects of no interest: drift, head motion, physiological variables, censored time points, tasks of no interest, etc.

Applying to experiment types Straightforward for resting state experiment: default mode

network (DMN) With tasks: correlation under specific condition(s) or resting

state? Program: 3dfim+ or 3dDeconvolve

Original regression: y = X + (t) New model: y = [X S(t)] + (t) r: linear correlation; slope for standardized Y and X β: slope, amount of linear change in Y when X increases by 1 unit

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Simple Correlation Analysis Group analysis

Run Fisher-transformation of r to Z-score and t-test: 3dttest Take β and run t-test (pseudo random-effects analysis): 3dttest

Take β + t-statistic and run random-effects model: 3dMEMA Caveats: don’t over-interpret

Correlation does not necessarily mean causation: no proof for anatomical connectivity (e.g., more than two regions in a network)

No golden standard procedure and so many versions in analysis: seed region selection, covariates, r (Z)/β, bandpass filtering, …

Measurement error problem: underestimation, attenuated bias

Website: http://afni.nimh.nih.gov/sscc/gangc/SimCorrAna.html Interactive tools in AFNI and SUMA: InstaCor, GroupInstaCor

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Context-Dependent Correlation Popular name: Psycho-Physiological Interaction (PPI)

3 explanatory variables Condition (or contrast) effect: C(t) Seed effect on rest of brain: S(t) Interaction between seed and condition (or contrast): I(C(t),

S(t))o Directionality here!

Model for each subject Original regression: y(t) = [C(t) Others]+(t) New model: y(t) = [C(t) S(t) I(C(t), S(t)) Others]+(t) 2 more regressors than original model: S(t), I(C(t), S(t)) Should effects of no interest be included in the model?

o Others NOT included in SPM What we care for: r or β for I(C(t), S(t)) I(C(t), S(t)): the variability in addition to C(t) and S(t)

Symmetrical modulation

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Seed

Target

Condition

PsychologicalPhysiological

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Context-Dependent Correlation How to formulate I(C(t), S(t))?

Interaction occurs at neuronal, not BOLD (an indirect measure) level Deconvolution: derive “neuronal response” at seed based on BOLD

responseo 3dTfitter: Impulse Neuronal events = BOLD response; Gamma NE(t) =

S(t)o Deconvolution matters more for event-related than block experiments

Interaction at neuronal level – 3dcalc: NE(t) × C(t) = NI(t)o Useful tool for C(t): timing_tool.py converts stimulus timing into 0s and 1s

Interaction at BOLD level - convolution – waver: Gamma NI(t) = I(C(t), S(t))

If stimuli were presented in a resolution finer than TR – not TR-lockedo 1dUpsample n: interpolate S(t) n finer before deconvolution 3dTffitero Downsample interaction I(C(t), S(t)) back to original TR: 1dcat with selector

'{0..$(n)}' Solving y(t) = [C(t) S(t) I(C(t), S(t)) Others]+(t) – 3dDeconvolve

Group analysis Run Fisher-transformation of r to Z-score and t-test: 3dttest Take β (+t): 3dttest (3dMEMA)

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PPI Caveats No proof for anatomical connectivity

Correlation does not necessarily mean causation If other regions involved in the network

Measurement error in regression Noisy seed time series Neuronal response hard to decode: Deconvolution is very far from

reliable, and we have to assume a shape-fixed HRF, same shape across conditions/regions/subjects

The errors lead to attenuation or regression dilution Doesn’t say anything about interaction between condition and

target on seed Doesn’t differentiate whether modulation is

Condition on neuronal connectivity from seed to target, or Neuronal connectivity from seed to target on condition effect

Website: http://afni.nimh.nih.gov/sscc/gangc/CD-CorrAna.html

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Seed

Target

Condition

PsychologicalPhysiological

Network-Based Modeling: a toy example A network with two regions: both

contemporaneous and delayed Within-region effects: lagged correlation Cross-regions effects: both instantaneous and lagged

If we have time series data from the two regions Can we evaluate the above model? Estimate and make inferences about the α values?

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Structure Equation Modeling (SEM): a toy example A network with two regions: no delayed effects

No within-region effects: no lagged effects – no temporal correlation!

Cross-regions effects: instantaneous correlation only; no lagged effects

If we have time series data from the two regions Can we evaluate the above model? Estimate and make inferences about the α values?

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Vector Autoregressive (VAR) Modeling: a toy example A network with two regions: no contemporaneous

effects Within-region effects: lagged effects Cross-regions effects: lagged effects only; no

instantaneous effects

If we have time series data from the two regions Can we evaluate the above model? Estimate and make inferences about the α values?

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Structure Equation Modeling (SEM) or Path Analysis General model for a network of n regions

Only consider instantaneous effects; assumes no delayed effects Data centered around mean; if possible, remove all confounding

effects Parameters in A0 code for cross-region path strength; zero

diagonals ε(t) ~ N(0, Ψ), Ψ: diagonal matrix (interregional correlations:

A0) Solving SEM

Compare covariance matrix from data with the one from the model

One problem: we can’t solve SEM if all parameters in A0 are unknown!o Totally n(n+1)/2 simultaneous equations; n(n-1)+n=n2 unknowns!o Can only allow at most n(n-1)/2 paths, half of the off-diagonalso Have to fix the rest paths (at least n(n-1)/2) to 0 or known values

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SEM: Model Validation Null hypothesis H0: It’s a good model about

instantaneous network Knowing directional connectivity btw ROIs, does data

support model? Want to see model (H0) not rejected

o χ2(n(n-1)/2-k)-test: badness-of-fito Fit indices (AIC, CFI, GFI, ): balance between optimization and model

complexity Input: model specification, covariance/correlation matrix,

etc. If H0 is not rejected, what are the path strengths?

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SEM: Model Comparison and Search

Comparing two nested models through χ2(1)-test For example, not sure about a pth

Search all possible models Sounds appealing: often seen in literature Problematic: data-driven vs. theory-based Learn from data, and don’t let data be your teacher

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SEM: More Serious Problems Correlations as input in SEM: popular practice

Usually practiced in social science studies for scaling issues Save DFs in FMRI data analysis Path coefficients not interpretable Can’t make statistical inferences: t-stat and CI, if provided,

are incorrect Assumptions

Within-region temporal correlations ignored Cross-regions: delayed interactions ignored

Data preprocessing Have to remove all confounding effects

Individual subjects vs. group How to combine multiple multiple subjects Fixed vs. random-effects analysis

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Vector Autoregression (VAR) General model for a network of n regions VAR(p)

X(t) = A1X(t-1)+…+ApX(t-p)+c1z1(t)+ …+cqzq (t)+ε(t) Only focus on lagged effects: Current state depends linearly on history Instantaneous effects modeled, but left in residuals as effects of no

interest Confounding (exogenous) effects can be incorporated as part of the

modelo Slow drift, head motion, physiological confounds, time breaks, conditions of no

interesto Unlike SEM, only minimal pre-processing needed (slice timing + motion correction)

Parameters in Ai code for cross-region path strengtho Meaning of path coefficients

Assumptionso Linearity; Stationarity/invariance: mean, variance, and auto-covarianceo ε(t) ~ N(0, Ψ), Ψ: not diagonal matrix (positive definite contemporaneous covariance);

no serial correlation in individual residual time series Rationale for VAR(p)

Response to stimuli does not occur simultaneously across brain: latency However, is data time resolution fine enough with TR = 1-2 sec???

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Solving VAR Model X(t) = A1X(t-1)+…+ApX(t-p)+c1z1(t)+ …+cqzq (t)+ε(t) Order selection with 4 criteria (1st two tend to overestimate)

o AIC: Akaike Information Criteriono FPE: Final Prediction Erroro HQ: Hannan-Quinn o SC: Schwartz Criterion

Solve VAR with OLS No need to specify connections as in SEM Obtain estimates of all elements in Ai, and make statistical

inferences based on t-statistic for each path Data driven instead of model validation? Model tuning when some covariates are not significant VAR as a seed-based analysis Bivariate autogression: use seed to search for regions that

may form a network with the seed 3dGC (vs. 1dGC)

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VAR Model Quality Check Stationarity: VAR(p) Y(t) = α+A1Y(t-1)+…+ApY(t-p)

+ε(t) Check characteristic polynomial det(In-A1z-…-Apzp)≠0 for |z|≤1

Residuals normality test Gaussian process: Jarque-Bera test (dependent on variable order) Skewness (symmetric or tilted?) Kurtosis (leptokurtic or spread-out?)

Residual autocorrelation Portmanteau test (asymptotic and adjusted) Breusch-Godfrey LM test Edgerton-Shukur F test

Autoregressive conditional heteroskedasticity (ARCH) Time-varying volatility

Structural stability/stationarity detection Is there any structural change in the data? Based on residuals or path coefficients

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VAR: Serious Problems Data sampling rate

Cross-region interactions occur probably at ms level, but usually TR = 1~2 seconds in FMRI time series (TR could be 100-200 ms with single-slice scanning)

Will VAR(1) catch the real lagged effects across regions???

With coarse sampling, the instantaneous effects will more likely reveal the real network than the lagged effects

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Network-Based Modeling: a toy example A network with two regions: both

contemporaneous and delayed Within-region effects: lagged correlation Cross-regions effects: both instantaneous and lagged

If we have time series data from the two regions Can we evaluate the above model? Estimate and make inferences about the α values?

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One World United Under One Flag! Why don’t we just combine SEM and VAR?

No reason we shouldn’t or cannot It’s called Structural Vector Autoregression (SVAR)! It accounts for variability from both instantaneous and lagged

effects It improves model quality and statistical power

General SVAR(p) model AB model: X(t)=A0X(t)+A1X(t-1)+…+ApX(t-p)+c1z1(t)+…+cqzq (t)

+Bε(t) A0 represents the cross-region instantaneous effects

o Diagonals are 0 Ai represents both within-region and cross-region lagged

effects B is a diagonal matrix so that ε(t) ~ N(0, I)

o All the cross-region instantaneous effects are contained in A0

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Solving SVAR AB model: X(t)=A0X(t)+A1X(t-1)+…+ApX(t-p)+c1z1(t)+…+cqzq (t)

+Bε(t) Equivalence to a reduced VAR(p) model

Ai* = (I-A0)-1Ai, cj*=(I-A0)-1cj, *(t) = (I-A0)-1Bε(t) Solve the reduced VAR(p), obtain estimates of Ai*, cj*, and

residual covariance * Solve (I-A0)-1BB(I-A0)-T = * through ML. Similar to SEM:

o Totally n(n+1)/2 simultaneous equations; n(n-1)+n=n2 unknowns!o Can only allow at most n(n-1)/2 paths in A0, half of the off-diagonalso Have to fix the rest paths (at least n(n-1)/2) to 0 or known valueso Model validation, comparison, and search for the instantaneous

network A0 Finally update Ai (and cj) for the lagged effects

AFNI program 1dSVAR

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What can we do with 1dSVAR If time resolution is too coarse (e.g., FMRI): Model

validation/comparison/search of the instantaneous network while accounting for the lagged effects Knowing directional connectivity btw ROIs, does data support model? Want to see model (H0) not rejected

o χ2(n(n-1)/2-k)-test: badness-of-fito Fit indices (AIC, CFI, GFI, ): balance between optimization and model complexity

If H0 is not rejected, what are the path strengths? If time resolution is good (e.g., MEG/EEG)

Both instantaneous and lagged effects are of interest? SEM+VAR

Lagged effects: data-driven; safe but inefficient (over-fitting) Instantaneous effects: theory/hypothesis-based; powerful but risky Various possibilities: e.g., borrow DFs for instantaneous effects from lagged effects?

Group analysis: MEMA

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SVAR: caveatso Assumptions (stationarity, linearity, Gaussian residuals, no

serial correlations in residuals, etc.)o Accurate ROI selection: If an essential region is missing

o Sensitive to lagso Confounding latency due to HDR variability and vascular

confoundso Model comparison/search

o Learn from data, but don’t let data be your teacher!

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SVAR applied to FMRI Resting state

Ideal situation: no cut and paste involved Physiological data maybe essential?

Block experiments Duration ≥ 5 seconds? Extraction via cut and paste

o Important especially when handling confounding effectso Tricky: where to cut especially when blocks not well-

separated? Event-related design

With rapid event-related, might not need to cut and paste (at least impractical)

Other tasks/conditions as confounding effects

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SVAR: Why not Granger Causality Causality: philosophical and

physiological/anatomical; effective? Granger causality: A Granger causes B if time series at

A provides statistically significant information about time series at B at some time delays (order) Causes must temporally precede effects Causality can be inferred from an F- or 2-test that shows the amount

of variability of overall lagged effects each connection accounts for Both instantaneous and lagged effects are

modeled in SVAR

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Network-based Analysis in AFNI Exploratory: ROI searching with 3dGC

Seed vs. rest of brain Bivariate model 3 paths: seed to target, target to seed, and self-effect Group analysis with 3dMEMA or 3dttest

Path strength significance testing in network: 1dSVAR Pre-selected ROIs SVAR model Multiple comparisons issue Group analysis

o path coefficients onlyo path coefficients + standard erroro F-statistic (BrainVoyager)

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